Potential Driven Flows Through Bifurcating Networks

Aerospace and Mechanical Engineering Graduate Student Conference 200619 October, 2006

Jason MayesAdvisor: Dr. Mihir Sen

Outline Background/Motivation Objectives A self-similar model Simplifying the model Forms of similarity Examples Conclusions / Future work

Background / Motivationself-similarity

Natural physical examples Broccoli

Artificially occurring examples Artificial terrain

Non-physical examples Data series

Music

"When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar." - Mandelbrot

From Hofstadter's classic, Godel, Escher, Bach: "A fugue is like a canon, in that it is usually based on one theme which gets played in different voices and different keys, and occasionally at different speeds or upside down or backwards."

Background / Motivationself-similar systems

Self-similarity in engineering many systems can be considered self-similar

over several scales Large scale problems

as systems grow, solutions become more difficult

can we simplify?

Objectivessimplification and reduction

Take advantage of similarity to simplify analysis

Use known structure / behavior to simplify Using structure Extending behavior from one scale to another

Reduce large equation sets

A self-similar modela model for analysis

The bifurcating tree geometry Geometry seen in a wide variety of

applications Potential-driven flow or transfer

ex. heat, fluid, energy, ect. Conservation at bifurcation points

q

q1

q2q = q1+q2

A self-similar modelpotential-driven transfer

Assumptions Transfer governed by a linear operator i.e., for each branch: pLq

A self-similar modelsimplification

Goal is to reduce or simplify the system to the single equation

Generation two (N=2) Network

A self-similar modelsimplification

Apply recursively to eliminate q1,1 and q2,2

A self-similar modela simple result

Result of simplification for N=2 network

For the more general N-generation network

For integro-differential operators Process repeated in the Laplace domain Inverses become simple algebraic inverses Can be written as a continued fraction

Forms of similaritywithin and between

Similarity can be used to further simplify Two forms of similarity:

Similarity ‘within’ a generation Symmetric networks: the operators within a generation are identical Asymmetric networks: the operators within a generation are not identical

Similarity ‘between’ generations Generation dependent operators depend on the generation in which the

operator occurs and change between successive generations Generation independent operators do not change between generations

Four possible combinations Symmetric with generation independent operators Asymmetric with generation independent operators Symmetric with generation dependent operators Asymmetric with generation independent operators

Examplestree of resistors

Symmetric with generation independent operators

Examplestree of resistors

Tree constructed of identical resistors Current i(t) through each branch is driven by the potential

difference v(t) across the branch Each branch governed by

For an N generational tree of resistors

For an infinite tree of resistors

)()( tvtRi )(

)(tvp

tiqRL

)()(211 tvtiR

N

)()( tvtRi

Examplesfractional order visco-elasticity models

Asymmetric with generation independent operators

Examplesfractional order visco-elasticity models

Tree is composed of springs and dashpots Branches governed by linear operators Result is a fractional-order visco-elasticity

modelkLk a

dtdL

dtd

a

Exampleslaminar pipe flows in branching networks

Symmetric with generation dependent operators

Exampleslaminar pipe flows in branching networks

Each pipe governed by

In Laplace domain

In the time domain

Conclusions / Future Work Known behavior on one scale can be extended

to better understand behavior on another Similarity or structure can be used to help

simplify For equation sets: patterns or structures can

be used to simplify Future work:

Analyze other self-similar geometries Study probabilistically self-similar geometries

Questions?